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Magma
magma: G := TransitiveGroup(16, 23);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $Q_8 : C_2^2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,16)(2,15)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13), (1,5)(2,6)(3,15)(4,16)(7,12)(8,11)(9,14)(10,13), (3,6)(4,5)(11,14)(12,13), (1,2)(3,5)(4,6)(7,9)(8,10)(11,12)(13,14)(15,16), (1,10)(2,9)(3,11)(4,12)(5,13)(6,14)(7,16)(8,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 15 $4$: $C_2^2$ x 35 $8$: $C_2^3$ x 15 $16$: $C_2^4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7
Degree 8: $C_2^3$, $Q_8:C_2^2$ x 2
Low degree siblings
8T22 x 6, 16T23 x 8, 32T9Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$1^{16}$ | $1$ | $1$ | $()$ | |
$2^{4},1^{8}$ | $2$ | $2$ | $( 3, 6)( 4, 5)(11,14)(12,13)$ | |
$2^{8}$ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$ | |
$2^{8}$ | $2$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,12)(13,14)(15,16)$ | |
$2^{8}$ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,11)( 8,12)( 9,13)(10,14)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 3,16, 6)( 2, 4,15, 5)( 7,11,10,14)( 8,12, 9,13)$ | |
$2^{8}$ | $2$ | $2$ | $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 4,16, 5)( 2, 3,15, 6)( 7,13,10,12)( 8,14, 9,11)$ | |
$2^{8}$ | $2$ | $2$ | $( 1, 7)( 2, 8)( 3,11)( 4,12)( 5,13)( 6,14)( 9,15)(10,16)$ | |
$2^{8}$ | $2$ | $2$ | $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,12)( 6,11)( 9,15)(10,16)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 8,16, 9)( 2, 7,15,10)( 3,12, 6,13)( 4,11, 5,14)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 8,16, 9)( 2, 7,15,10)( 3,13, 6,12)( 4,14, 5,11)$ | |
$2^{8}$ | $2$ | $2$ | $( 1,11)( 2,12)( 3, 7)( 4, 8)( 5, 9)( 6,10)(13,15)(14,16)$ | |
$4^{4}$ | $2$ | $4$ | $( 1,11,16,14)( 2,12,15,13)( 3,10, 6, 7)( 4, 9, 5, 8)$ | |
$4^{4}$ | $2$ | $4$ | $( 1,12,16,13)( 2,11,15,14)( 3, 8, 6, 9)( 4, 7, 5,10)$ | |
$2^{8}$ | $2$ | $2$ | $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,16)(14,15)$ | |
$2^{8}$ | $1$ | $2$ | $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $32=2^{5}$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $2$ | ||
Label: | 32.49 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | ||
Size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2A | |
Type | ||||||||||||||||||
32.49.1a | R | |||||||||||||||||
32.49.1b | R | |||||||||||||||||
32.49.1c | R | |||||||||||||||||
32.49.1d | R | |||||||||||||||||
32.49.1e | R | |||||||||||||||||
32.49.1f | R | |||||||||||||||||
32.49.1g | R | |||||||||||||||||
32.49.1h | R | |||||||||||||||||
32.49.1i | R | |||||||||||||||||
32.49.1j | R | |||||||||||||||||
32.49.1k | R | |||||||||||||||||
32.49.1l | R | |||||||||||||||||
32.49.1m | R | |||||||||||||||||
32.49.1n | R | |||||||||||||||||
32.49.1o | R | |||||||||||||||||
32.49.1p | R | |||||||||||||||||
32.49.4a | R |
magma: CharacterTable(G);